Method of Precoding with a Codebook for a Wireless System

ABSTRACT

A quantized multi-rank beamforming scheme for multiple-antenna systems such as a multiple-input-multiple-output (MIMO) wireless downlink. User equipment (UE) estimates downlink channel and transmit power and determines rank and power allocations. A quantized beamforming matrix is then determined by the UE using successive beamforming. The UE also determines channel quality indices (CQI) which it feeds-back to the wireless downlink base station along with the index of the quantized beamforming matrix. The base station uses the CQI information to select a UE for scheduling of downlink transmission and the quantized beamforming matrix index received from the selected UE to beamform the downlink transmission to the UE. Base station overhead and is minimized while providing near-optimal performance given the constraints of a limited feed-back channel and computational complexity of the UE.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a division of co-pending patent application Ser. No.13/029,561, entitled “Method of Precoding with a Codebook for a WirelessSystem With Multiple Transmission Ranks and a Quantized BeamformingMatrix,” filed 17 Feb. 2011, which in turn is a division of patentapplication Ser. No. 11/674,330, entitled “Structured Codebook andSuccessive Beamforming for Multiple Antenna Systems”, filed 13 Feb.2007, which in turn claimed priority to Provisional Application No.60/743,290, filed 14 Feb. 2006 and claimed the benefit of patentapplication Ser. No. 11/554,278, filed 30 Oct. 2006, the entire contentsand file wrappers of which are hereby incorporated by reference for allpurposes into this application.

FIELD OF THE INVENTION

The present invention relates to the field of wireless communications,particularly wireless, high-rate communications using multiple-antennasystems.

BACKGROUND INFORMATION

The hostility of the wireless fading environment and channel variationmakes the design of high rate communication systems very challenging. Tothis end, multiple-antenna systems have shown to be effective in fadingenvironments by providing significant performance improvements andachievable data rates in comparison to single antenna systems. Wirelesscommunication systems employing multiple antennas both at thetransmitter and the receiver demonstrate tremendous potential to meetthe spectral efficiency requirements for next generation wirelessapplications. This has spurred research in the efficient design anddeployment of various multiple-input-multiple-output (MIMO)configurations for practical systems. The collection of papers in IEEETransactions on Information Theory, vol. 49, Issue 10, October 2003,represents a sample of the wide array of research in MIMO systems.

Moreover, multiple transmit and receive antennas have become an integralpart of the standards of many wireless systems such as cellular systemsand wireless LANs. In particular, the recent development of UMTSTerrestrial Radio Access Network (UTRAN) and Evolved-UTRA has raised theneed for multiple antenna systems to reach higher user data rates andbetter quality of service, thereby resulting in an improved overallthroughput and better coverage. A number of proposals have discussed andconcluded the need for multiple antenna systems to achieve the targetspectral efficiency, throughput, and reliability of EUTRA. Theseproposals have considered different modes of operation applicable todifferent scenarios. The basic assumptions that vary among suchproposals include: (i) using a single stream versus multiple streams;(ii) scheduling one user at a time versus multiple users; (iii) havingmultiple streams per user versus a single stream per user; and (iv)coding across multiple streams versus using independent streams. A basiccommon factor, however, among the various downlink physical layer MIMOproposals is a feedback strategy to control the transmission rate andpossibly a variation in transmission strategy.

While the proposals for the use of multiple antenna systems in downlinkEUTRA such as per antenna rate control (PARC), per stream rate control(PSRC), per group rate control (PGRC), per user and stream rate control(PUSRC), per user unitary rate control (PU2RC), single codeword/multiplecodeword transmission (SCW/MCW), spatial domain multiplex/spatial domainmultiple access (SDM/SDMA), and current transmit diversity scheme inrelease 6 such as selection transmit diversity (STD), space-timetransmit diversity (STTD), and transmit adaptive antennas (TxAA) differin terms of the system description, they all share the followingfeatures: (i) possible multiplexing of streams to multiple streams; (ii)possible use of linear precoding of streams before sending to antennas;(iii) possible layering of the streams between the antennas; and (iv)rate control per stream or multiple jointly coded streams.

The performance gain achieved by multiple antenna system increases whenthe knowledge of the channel state information (CSI) at each end, eitherthe receiver or transmitter, is increased. Although perfect CSI isdesirable, practical systems are usually built only on estimating theCSI at the receiver, and possibly feeding back the CSI to thetransmitter through a feedback link with a very limited capacity. UsingCSI at the transmitter, the transmission strategy is adapted over space(multiple antennas) and over time (over multiple blocks).

The performance of multiple antenna systems with or without knowledge ofthe channel state information has been extensively analyzed over thelast decade. Partial feedback models have been considered due to thelimitations of the feedback channel from the receiver to thetransmitter. Different partial feedback models include: channel meanfeedback (see, e.g., A. Narula et al., “Efficient use of sideinformation in multiple-antenna data transmission over fading channels,”IEEE Journal on Selected Areas of Communications, vol. 16, no. 8, pp.1423-1436, October 1998); channel covariance feedback (E. Visotsky etal., “Space-time precoding with imperfect feedback,” in Proceedings ISIT2000, Sorrento, Italy, June 2000); feedback of k-out-of-min(M,N)eigenvectors and eigenvalues of an M×N multiple antenna channel (J. Rohet al., “Multiple antenna channels with partial channel stateinformation at the transmitter,” Wireless Communications, IEEETransactions on, vol. 3, pp. 677-688, 2004); partial feedback based onstatistical model and robust design (A. Abdel-Samad et al., “Robusttransmit eigen-beamforming based on imperfect channel stateinformation,” in Smart Antennas, 2004. ITG Workshop on, 2004); andquantized feedback.

Beamforming introduces an alternative use of quantized feedback bits inwhich the design is almost independent of the average received SNR andconstant average transmit power is assumed at the transmitter. Inbeamforming, independent streams are transmitted along differenteigenmodes of the channel resulting in high transmission rates withoutthe need to perform space-time coding.

Beamforming has received considerable attention for the case of multipletransmit antennas and a single receive antenna. (See, e.g., Narula etal. cited above.) Rate regions for the optimality of dominanteigen-beamformers (rank-one beamformers) in the sense of maximizingmutual information has also been studied (see, e.g., S. A. Jafar, etal., “Throughput maximization with multiple codes and partial outages,”in Proceedings of Globecom Conference, San Antonio, Tex., USA, 2001), ashave systematic constructions for finite-size beamformer codebooks formultiple transmit single receive antenna systems resulting in nearoptimal performance. Similarly, a design criterion for a dominanteigen-beamformer, for use with both single and multiple receive antennasystems, has been proposed. (See D. Love et al.,“Grassmannianbeamforming for multiple-input multiple-output wireless systems,” IEEETransactions on Information Theory, vol. 49, pp. 2735-2747, 2003.) Asdiscovered by the inventors of the present invention, however, such unitrank beamformers can result in significant performance degradation withMIMO systems for certain transmission rates, thus requiring higher ranktransmission schemes.

The design of higher rank beamformers for MIMO systems has also beenstudied in the past. Rob et al. addressed the problem of MIMObeamforming in the presence of perfect knowledge about a subset of thechannel eigenvectors. Knopp et al. explored the design of joint powercontrol and beamforming when the eigenvectors and the eigenvalues arecompletely known at the transmitter. (See R. Knopp et al., “Powercontrol and beamforming for systems with multiple transmit and receiveantennas,” IEEE Transactions on Wireless Communications, vol. 1, pp.638-648, October 2002.) Jafar et al. derived the conditions for theoptimality of MIMO beamforming (in the sense of achieving capacity) whenthe channel covariance is fed back to the transmitter. (See, S. A. Jafaret al. “Throughput maximization with multiple codes and partialoutages,” in Proceedings of Globecom Conference, San Antonio, Tex., USA,2001.) The design of MIMO systems using multiple simultaneous streamsfor transmission when finite rate feedback is available in the systemhas also been studied (see Love et al., cited above.) The designcriterion therein sought to quantize the set of active eigenvectors suchthat the loss in SNR compared to perfect channel feedback is minimized.

In addition to the aforementioned considerations, it is desirable toachieve the highest possible spectral efficiencies in MEMO systems withreasonable receiver and transmitter complexity. Though theoreticallyspace-time codes are capable of delivering very high spectralefficiencies, e.g. 100s of megabits per second, their implementationbecomes increasingly prohibitive as the bandwidth of the systemincreases.

SUMMARY OF THE INVENTION

The present invention is directed to quantized, multi-rank beamformingmethods and apparatus. Embodiments of the present invention canconsiderably outperform known beamforming and precoding techniques fordifferent transmission rates. Advantageously, the present invention canbe implemented with low complexity at the transmitter, and can operatewith a low feedback rate, common in most practical systems.

In an exemplary embodiment, a multi-rank beamforming strategy ispresented in order to efficiently use the available spatial diversity inthe MIMO downlink of Evolved-UTRA and UTRAN with the goal of achievinghigher user data rates, and better quality of service with improvedoverall throughput and better coverage. The exemplary embodimentcombines transmission rank control, rank-specific and structuredquantized precoding codebook design, and successive beamforming.

A structured codebook in accordance with the present invention allowsfor successive beamforming and considerably reduces the memoryrequirement and computational complexity of the algorithm in comparisonto the optimal codebook, while providing near-optimal performance.Successive beamforming in accordance with the present invention can beused to perform a finer quantization of a single vector in rank-1transmission. Furthermore, re-use of the same quantization codebook forall ranks leads to a significant reduction in the memory required tostore quantization vectors.

The aforementioned and other features and aspects of the presentinvention are described in greater detail below.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic block diagram of a multiple-antenna communicationssystem with quantized feedback of channel state information.

FIG. 2 is a block diagram of an exemplary embodiment of a base stationof an orthogonal frequency-division multiplexing (OFDM) wireless MIMOcommunication system with feedback from the user equipment (UE).

FIG. 3 is a flowchart of the operation of a base station, such as thatof FIG. 2, in accordance with an exemplary embodiment of the presentinvention.

FIG. 4 is a flowchart of the operation of a UE in accordance with anexemplary embodiment of the present invention.

FIG. 5 is a flowchart of an algorithm for determining rank and powerallocations for a downlink using a modified capacity measure inaccordance with an exemplary embodiment of the present invention.

FIG. 6 is a flowchart of a successive beamforming algorithm inaccordance with an exemplary embodiment of the present invention.

FIG. 7 is a flowchart of a CQI calculation algorithm in accordance withan exemplary embodiment of the present invention.

FIG. 8A is a table of exemplary parameters used in simulating anexemplary embodiment of the present invention and FIG. 8B shows anexemplary format used for feedback information in such an embodiment.

FIG. 9 is a graph of bits per chunk vs. chunk size for the simulatedexemplary embodiment of the present invention.

FIG. 10 is a graph of throughput vs. SNR for the simulated exemplaryscheme against fundamental limits using a finite set of rates (5 CQIbits).

FIG. 11 is a graph of throughput vs. SNR for the simulated exemplaryscheme as compared to selected known schemes.

FIG. 12 is a graph of throughput vs. SNR showing the effect of feedbackerror for the simulated exemplary embodiment of the present invention.

FIGS. 13A and 13B show a flowchart which illustrates the UE operationfor an alternative exemplary embodiment of a rank adaptation scheme fora MIMO downlink in which the beamforming rank may be provided by thebase station and which can accommodate various receivers.

FIG. 14 shows a flowchart of an algorithm for calculating CQIs for MCWtransmission for reception by a MMSE-SIC decoder.

FIG. 15 shows a flowchart of an algorithm for calculating a CQI for SCWtransmission for reception by a ML decoder.

FIG. 16 shows a flowchart of an algorithm for calculating a CQI for SCWtransmission for reception by a LMMSE decoder.

DETAILED DESCRIPTION

An exemplary multiple-antenna communication system 100 with quantizedfeedback is schematically shown in FIG. 1. A transmitter 110 transmitsfrom t transmitting antennas 111.1-111.t over a fading channel 130 to rreceiving antennas 121.1-121.r coupled to a receiver 120. A channelestimator 125 provides an estimate of the channel 130 to the receiver120. The channel estimate is also quantized and provided to thetransmitter 110 via a quantized rate control feedback channel 135.

For purposes of analysis, a flat fading channel model is assumed inwhich the channel remains constant for each block of transmission.Furthermore, the feedback channel 135 is assumed to be an error-free,zero-delay feedback channel from the receiver to the transmitter,carrying B bits of information about the channel realization everyframe.

For a multiple-antenna system with r receive and t transmit antennas thebaseband channel model can be expressed as follows:

Y=HX+W,  (1)

where Y is the r×1 received column vector, H is the r×t channel matrix,X is the t×1 transmit column vector, and W is the r×1 noise columnvector. The input is subject to an average power constraint P, i.e.:

tr(Q)≦P, where Q=

[XX ^(H)],  (2)

[.] denotes the expected value, and tr(.) represents the trace of amatrix. A goal of an exemplary space adaptation scheme is to minimizethe frame error rate, which tightly follows the outage behavior of thetransmission strategy, defined as:

P _(out)=Prob{log det(I _(n) +HQH ^(H))<R},

s.t. tr(Q)≦P, Q≧0  (3)

where I_(n) is an identity matrix of size n and R is the attemptedtransmission rate.

In an exemplary multi-rank beamforming scheme in accordance with thepresent invention, channel state information (CSI) is available to thetransmitter (CSIT) as well as the receiver (CSIR). Where perfect CSITand CSIR are assumed, the capacity of the multiple-antenna fadingchannel 130 can be achieved through power adaptation over time (oneaverage power for each channel state) and water-filling power controlover multiple eigenvectors of the channel for each block oftransmission. This translates into power control over multiple antennasin the spatial domain. The nature of the water-filling power controlimplies that transmission may occur only on a subset of eigenvectors ofthe channel depending on the channel condition. Therefore, to maximizethe system throughput, the number of eigenvectors in which communicationoccurs, defined as the transmission rank, is controlled. For a giventransmission rate, the transmission rank depends on the channelcondition, where the transmission rank is at most equal to m=min(t, r).Thus, the set of all channel conditions is divided into m partitions,where the k-th partition represents the channel conditions for which thebeamformer rank is equal to k, i.e., transmission occurs on the largestk eigenvectors out of a total of min(t, r) eigenvectors of the channelH.

Codebook Design

The codebook design for the beamforming or precoding in multiple antennasystem entails finding the best packing in a Grassmanian manifold.

In general, each precoding matrix U of size t×n defines an n-dimensionalsubspace of a t-dimensional complex vector space W=

′, where C is the space of complex numbers. The matrix U mayinterchangeably be used to denote the space that is spanned by U. Theset of all precoding matrices of size t×n constitutes a Grassmannianmanifold denoted by G(t, n) or G_(n)(W). The set of all beamformingvectors of size t×1 forms a Grassmanian manifold of G(t, 1), that isalso known as the projective space P(W).

A packing in a Grassmannian manifold is then defined with respect to ametric in the corresponding space. Using the metric, the distancebetween two points on the manifold can be defined. Since each point inGrassmanian manifold G(t, n) is a n-dimensional space, the metric infact measures the subspace distance between two t-dimensional subspaces.

Two metrics have been used for generating the precoding codebooks inmultiple antenna communication systems: chordal distance andFubini-Study distance. These metrics, however, are not the best metricsto be used for the design of precoders for multiple antenna systems. Wecan show that the packing with respect to the Fubini-Study metric givesthe precoding codebook that is good only at very high SNRs and thepacking with respect to Chordal distance gives the precoding codebookthat is good only at very low SNRs.

In an exemplary embodiment, a new metric (called “p-metric”) is usedwhich is provably a valid metric for all the positive real values of aparameter pε

. The p-metric enables the design of a codebook that is good for adesired range of SNR. An interesting property of the p-metric is that asp goes to infinity, the p-metric becomes equivalent to the Fubini-Studymetric and as p goes to zero, the p-metric becomes equivalent to chordaldistance. In other words, “∞-metric” and “0-metric” are equivalent toFubini-Study and chordal metric, respectively.

The chordal distance, Fubini-Study distance, and p-metric between twosubspaces V_(t×n) and U_(t×n) are respectively defined as:

$\begin{matrix}{{d_{chordal} = \left. \frac{1}{\sqrt{2}}||{{UU}^{*} - {VV}^{*}} \right.||_{F}},} & (4) \\{{d_{FS} = {{Arc}\mspace{14mu} {\cos \left( \left| {\det \left( {U^{*}V} \right)} \right| \right)}}},} & (5) \\{d_{P} = {{Arc}\mspace{14mu} {{\cos \left( \left| {\det \left( \frac{I + {{pU}^{*}{VV}^{*}U}}{\left( {1 + p} \right)^{n}} \right)} \right|^{\frac{1}{2}} \right)}.}}} & (6)\end{matrix}$

For the purpose of successive beamforming, one set of quantizedbeamforming codebooks is generated consisting of k codebooks, one foreach packing of lines in the Grassmannian manifold of G(t−i+1, 1), fori=1, 2, . . . , k. The optimal packing uses the p-metric as the measureof the distance between the subspaces where the parameter p is chosensome where in the desired range of SNR. The distribution of the vectorsis not necessarily isotropic, however, therefore the codebook designdepends on the channel statistics and is not necessarily the packingwith respect to this metric. For an iid Rayleigh channel, the optimalcodebook is generated by finding the optimal packing with respect to thep-metric. At high SNR, this packing becomes the packing with respect tothe Fubini-Study metric and at low SNR becomes the packing with respectto chordal distance.

Successive Beamforming

The singular value decomposition of the channel estimate H can beexpressed as follows:

H=UDV ^(•),  (7)

where U and V are unitary matrices representing the left and righteigenvectors of H and D is a diagonal matrix of eigenvalues indescending order (V^(•) denotes the hermitian of matrix V). The columnof the unitary matrix V represents different eigenmodes of the channel.

V=[v ₁ v ₂ . . . v _(n)].  (8)

An exemplary multi-rank beamformer picks the first k columns of thematrix V that correspond to the first k dominant eigenmodes of thechannel due to the properties of the singular value decomposition.Choosing the rank of the beamformer is based on a long term rankprediction policy by the transmitter, e.g. the base station (BS), or itcan be performed at the receiver, e.g. the mobile user equipment (UE),by using a modified capacity calculation. Details of exemplary schemesof performing rank adaptation are described below with reference toflowcharts shown in FIGS. 3-7 and 13A-16.

For a given rank k, the aim of successive beamforming is to find thebest quantization of the actual k dominant eigenmodes of the channel.The successive beamforming for [v₁ v₂ . . . v_(k)] is performed asfollows.

First, v₁ is quantized using the codebook

^((t)) of the t-vectors in t-dimensional complex vector space

′. The quantized vector u₁ε

^((t)) is chosen to maximize

v₁,u₁

=|v₁ ^(•)u₁|. The index of u₁ constitutes the quantized feedbackinformation for the first vector.

Second, a rotation matrix φ(u₁) is found such that:

φ(u ₁)u ₁ =e ₁=[1;0;0; . . . ;0]  (9)

where e₁ denotes a unit norm vector in a principal direction [1; 0; 0; .. . ; 0].

Third, all of the vectors v₁ v₂ . . . v_(n) are rotated by the rotationmatrix φ(u₁), whereby:

V′=[v ₁ ′ v ₂ ′ . . . v _(n)]=φ(u ₁)V=[φ(u ₁)v ₁ φ(u ₁)v ₂ . . . φ(u ₁)v_(n)],  (10)

where all of the first elements of v₂′ v₃′ . . . v_(n)′ are zero due tothe fact that V is a unitary matrix and all of its columns areorthogonal. Moreover, if u₁=v₁, the first vector v₁′ becomes e₁.

Fourth, the same beamforming procedure described in the first threesteps are applied to a new matrix:

{tilde over (V)}=V′(2:end,2:end).  (11)

This step is successively performed until all the vectors are quantized.

The above successive beamforming technique generates different amountsof feedback for different ranks. Therefore, if the number of feedbackbits is given, different size codebooks can be generated to be used fordifferent rank beamforming. For example, where the possible rank of thechannel is 1 or 2, and 8 bits of feedback are available, 1 bit can bededicated for rank selection, and the other 7 bits for description ofthe eigenmodes. For rank 1, the 7 bits would provide for 2⁷=128t-vectors to be used for the quantization of the dominant eigenmode ofthe channel. For rank 2, 4 bits can be used for the first eigenmode and3 bits for the second eigenmode. As such, there would be 2⁴=16 t-vectorsto quantize the first eigenmode, and 2³=8 (t−1)-vectors to quantize thesecond eigenmode.

Due to the computational complexity and memory requirement of such abeamforming strategy, an alternative exemplary beamforming strategywhich uses the same idea of successive beamforming described above toquantize the vectors for the rank-1 case, will now be described. In thisexemplary embodiment, the same codebook of 16 t-vectors and 8(t−1)-vectors that is used for rank-2 beamforming is also used forrank-1 beamforming. Assume that only one vector v₁ is to be quantizedusing k codebooks comprised of t-vectors, (t−1)-vectors, etc., up to andincluding (t−k+1)-vectors, respectively.

First, v₁ is quantized using the codebook

^((t)) of the t-vectors in

′. The quantized vector u₁ε

^((t)) is chosen to maximize

v₁u₁

=|v₁ ^(•)u₁|.

Second, to find a finer description of v₁, the residual part of v₁ thatlies in the orthogonal space defined by span{u₁ ^(⊥)} is determined.Let:

v ₂ =v ₁−(v ₁ ^(•) u ₁)u ₁,  (12)

which is then normalized:

v ₂ ′=v ₂ /|v ₂|.  (13)

A rotation matrix φ(u₁) is then determined such that:

φ(u ₁)u ₁ =e ₁=[1;0;0; . . . ;0].  (14)

Third, the vector v₂ is rotated by the rotation matrix φ(u₁):

v ₂″=φ(u ₁)v ₂′,  (15)

where the first element of v₂″ is zero due to the fact that V is aunitary matrix and all of its columns are orthogonal.

In a fourth step, the above three steps are then performed on a newvector, {tilde over (v)}₂=v₂″(2: end). This step will be performedsuccessively until all k codebooks of t-vectors, (t−1)-vectors, up to(t−k+1)-vectors are used.

Therefore the exemplary successive beamforming method is performed onv₁, its residual on the orthogonal space span{u₁ ^(⊥)} defined by v₂″,and so on, instead of the orthogonal modes v₁, v₂, . . . . By thusapplying successive beamforming to the residual vectors, the ratio ofthe projection of v₁ in each successive space needs to be quantized forthe reconstruction. For example, for k=2, this entails quantizing thevalue |v₁ ^(•)u₁|/|v₂|. In some communication systems, feedback bitsthat are reserved for the feedback of the rate information for therank-2 transmission strategy may be used to convey the quantization ofthis value.

Exemplary Codebook Representation

The exemplary precoder selection process described above relies on a setof vectors V¹={v₁ ¹εC^(M)}_(i=1) ^(N) ¹ , V²={v_(i) ²εC^(M-1)}_(i=1)^(N) ² , . . . , V^(M-1)={v_(i) ^(M-1)εC²}_(i=1) ^(N) ^(M-1) , whereC^(N) denotes the N-dimensional complex space and a set of rotationsdefined by φ(v) for all vectors v in the codebook.

An exemplary representation of the codebook based on extending a resultfrom the real vector space to the complex vector space will now bedescribed. It is known that any M×M unitary matrix VεR^(M)×R^(M) (whereR denotes the set of real numbers) can be written as:

$\begin{matrix}{{V = \left\lbrack {v^{1},{{{HH}\left( {v^{1} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\v^{2}\end{bmatrix}},{{{HH}\left( {v^{1} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\{{{HH}\left( {v^{2} - e_{1}^{M - 1}} \right)}\begin{bmatrix}0 \\v^{3}\end{bmatrix}}\end{bmatrix}}} \right\rbrack},\ldots} & (16)\end{matrix}$

where v^(a) denotes the a^(th) column of matrix V and v_(b) ^(a) denotesthe b^(th) element of the vector v^(a) and

${H(w)} = {I - {2\frac{w^{T}}{\left. ||w \right.||^{2}}}}$

is the Householder transformation. This expansion, however, cannot begenerally done for a matrix VεC^(M)×C^(M), where C is the set of complexnumbers.

Any Unitary matrix V can be written in the form of:

$\begin{matrix}{{V = \left\lbrack {v^{1},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\v^{2}\end{bmatrix}}},{\Phi \; \left( v_{1}^{1} \right){{{HH}\left( {v^{1} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\{{\Phi \left( v_{2}^{2} \right)}{{{HH}\left( {v^{2} - e_{1}^{M - 1}} \right)}\begin{bmatrix}0 \\v^{3}\end{bmatrix}}}\end{bmatrix}}}} \right\rbrack},\ldots,} & (17)\end{matrix}$

where e₁ ^(N)=[1, 0, . . . , 0]^(T)εC^(N), v^(a) denotes the a^(th)column of matrix V, v_(b) ^(a) denotes the b^(th) element of the vectorv^(a), and the function Φ(v_(b) ^(a)), called the phase function, isdefined as

${\Phi \left( v_{b}^{a} \right)} = {\frac{v_{b}^{a}}{\left. ||v_{b}^{a} \right.||}.}$

Based on the above expansion, the codebook for a MIMO system with Mantennas is defined by sets of:

V ¹ ={v _(i) ¹ εC ^(M)}_(i=1) ^(N) ¹ , V ² ={v _(i) ² εC ^(M-1)}_(i=1)^(N) ² , . . . , V ^(M-1) ={v _(i) ^(M-1) εC ²}_(i=1) ^(N) ^(M-1),  (18)

where the first elements of all the vectors are real-valued. The set ofphases Φ(v_(b) ^(a)) can be used in the design of the codebook based onthe channel characteristics. The values of Φ(v_(b) ^(a)), however, donot directly affect the computation of the inner product in finding theprecoding matrix index.

In an exemplary case, all phase functions are equal to one. In thiscase, the set of precoding matrices are formed using these vectors alongwith the unitary Householder matrices of the form,

${H(w)} = {I - {2\frac{{ww}^{*}}{\left. ||w \right.||^{2}}}}$

(which is completely determined by the vector w). Then, for instance, arank-3 codeword can be constructed from three vectors v_(i) ¹εV¹, v_(j)²εV², v_(k) ³εV³ as:

$\begin{matrix}{{A\left( {v_{i}^{1},v_{j}^{2},v_{k}^{3}} \right)} = {\left\lbrack {v_{i}^{1},{{{HH}\left( {v_{i}^{1} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\v_{j}^{2}\end{bmatrix}},{{{HH}\left( {v_{i}^{1} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\{{{HH}\left( {v_{j}^{2} - e_{1}^{M - 1}} \right)}\begin{bmatrix}0 \\v_{k}^{3}\end{bmatrix}}\end{bmatrix}}} \right\rbrack.}} & (19)\end{matrix}$

An advantageous aspect of the exemplary scheme is that only the set ofvectors V¹, . . . , V^(M-1) along with some complex scalars are storedat the UE which results in a considerably lower memory requirementcompared to unstructured matrix codebooks.

The matrix representation of the codebook can be stored at the basestation, where memory requirements are not as stringent. For a givenchannel realization, the UE does not have to construct the precodingmatrix to determine the optimal precoder index and the correspondingMMSE filter.

For example, if the base station has M=4 transmit antennas, for acodebook of size 16 (per-rank), the following vector codebooks can beconstructed:

V ¹ ={v _(i) ¹ εC ⁴}_(i=1) ⁴ , V ² ={v _(j) ² εC ³}_(j=1) ⁴ , V³=[1,0]^(T) εC ².  (20)

For convenience, the case of a 2-antenna UE is considered. Thegeneralization to the 4-antenna case is straightforward. The codebook ofrank-2 contains 16 precoding matrices obtained as:

$\begin{matrix}{{{A\left( {v_{i}^{1},v_{j}^{2}} \right)} = \left\lbrack {v_{i}^{1},{{{HH}\left( {v_{i}^{1} - e_{1}^{4}} \right)}\begin{bmatrix}0 \\v_{j}^{2}\end{bmatrix}}} \right\rbrack},{1 \leq i},{j \leq 4},} & (21)\end{matrix}$

and the codebook of rank-1 also has 16 possibilities that are obtainedas the second columns of all possible {A(v_(i) ¹,v_(j) ²)},respectively.

Rank Adaptation Scheme for MIMO Downlink

FIG. 2 is a block diagram of the downlink portion of an exemplaryembodiment of a base station (BS) 200 of an orthogonalfrequency-division multiplexing (OFDM) wireless MIMO communicationsystem with feedback from the user equipment (UE). The base station 200comprises a downlink scheduler 210, a multiplexer 220, multiple AdaptiveModulation and Coding scheme blocks (AMC) 230.1-230.k, and a beamformingblock 250 driving multiple transmit antennas 260.1-260.t. Thebeamforming block 250 comprises power controllers 255.1-255.k, whichscale the power of the signals for each stream to be transmitted along kdifferent eigenvectors 265.1-265.k which are then combined in 270 andtransmitted by the transmit antennas 260.1-260.t.

The various blocks of the base station 200 operate in accordance withinformation fed-back from UE (not shown), including, for example, rank,beamforming matrix index, quantization power control, and Signal toInterference and Noise Ratio (SINR). The SINR information fed-back fromthe UEs is used by the downlink scheduler 210 to select a user streamfrom a plurality of user streams for transmission over the downlink tothe intended UE. Based on the rank feedback, the multiplexer block 220generates the appropriate number of signal streams and the AMC blocks230 choose the corresponding modulation and coding for each stream.

In an exemplary embodiment of the present invention, in order to achievethe best performance over an OFDM-based downlink with reasonablefeedback, the available sub-carriers are divided into chunks of adjacenttones and provide a feedback signal including the rank information, thebeamforming matrix index, and channel quality indices (CQIs) on aper-chunk basis. The CQIs can be the SINR for each stream. A chunk sizeof one is throughput optimal. Simulation results show, however, that alarger chunk size (if chosen properly) can significantly reduce thefeedback overhead with almost negligible loss in throughput. This factfollows the property that the precoder, i.e., beamforming matrix, chosenfor a tone usually is also the best precoder for the adjacent tones (upto a certain neighborhood size) out of the available quantized precodersin the codebook. It is possible to optimally design and also selectwideband precoders for the set of parallel channels, i.e., all tones inthe chunk. An exemplary strategy of selecting the optimal precoder forthe center tone in each chunk and then using it for the entire chunk isdescribed below. Moreover, to further reduce the feedback from the UEs,each UE can choose to send information about only the first few of its“best” chunks rather than all of them.

FIG. 3 is a flowchart which generally illustrates the operation of abase station (BS) in accordance with an exemplary embodiment of thepresent invention. FIG. 4 is a flowchart illustrating the operation of aUE in accordance with the exemplary embodiment. A high-level descriptionof FIGS. 3 and 4 is provided and details such as particular chunk-sizeused, OFDM implementation, etc. have been omitted for convenience.

As shown in FIG. 3, at step 310, the base station obtains the variousitems of feedback information from each UE, including the rankinformation (k), the beamforming matrix index (i), and CQI(s). At 320,the base station selects the UE with the highest supportable rate and at330, generates signal streams for the selected UE according to its rankand CQI(s). The signal of each stream is scaled at 340 in accordancewith a power allocation ratio and at 350, each stream is multiplied bythe corresponding column of the precoding matrix U and transmitted viathe multiple transmit antennas to the receiving UE.

At the UE, as shown in FIG. 4, the channel matrix H, the transmit power,and noise variance are estimated at 410. The ratio of transmit power tonoise variance defines the transmit SNR used in the selection of therank and the beamforming matrix. At step 420, the UE determines the rankk and power allocation P_(i) using a modified capacity measure. In analternative embodiment described below, these parameters are determinedby the base station. An algorithm for carrying out step 420 is describedbelow in greater detail with reference to FIG. 5.

At 430, the UE determines the precoding matrix U_(i) using successivebeamforming. An exemplary successive beamforming algorithm which allowsfor a considerable reduction in computational complexity as well as thememory requirement of the UE without sacrificing much throughputperformance is described below with reference to FIG. 6.

At 440, the UE finds the index of the augmented precoding matrix asdetermined by the precoding matrix U_(i) and the power allocation P_(i).

At 450, the one or more CQIs are then calculated for the correspondingreceiver (e.g., LMMSE or MMSE-SIC) using multiple or single codewordtransmission. A CQI calculation algorithm is described below withreference to FIG. 7.

FIG. 5 is a flowchart of an algorithm for determining the rank k andpower allocations P_(i) using a modified capacity measure f(H), definedbelow. At 510, the estimated channel matrix H, transmit power P, andnoise variance are provided as inputs. At 520, the singular valuedecomposition is performed, as described above. At 530 and 540, aprocessing loop is initiated starting with the highest possible rank. At550-570, for all quantized power vectors in the codebook of a givenrank, the modified capacity measure F_(i) is calculated in accordancewith the following expression:

f(H)=f(UDV*)=F _(i)=log det(I+PD ² diag(P _(i))).  (22)

where H is the estimated channel matrix and H=UDV* is the singular valuedecomposition of H.

At 580, the maximum value of the modified capacity measure F for thegiven rank is selected and designated F^((k)), referred to as the sumrate. At 590, a determination is made as whether the highest rank hasbeen processed or whether F^((k)) is increasing. If either condition istrue, operation branches to 591, in which the rank is decremented andthe loop starting at 540 is repeated. If not, operation proceeds to 592in which the rank and power allocation that maximize the sum rateF^((k)) are provided as outputs.

In an alternative embodiment, successive beamforming can be performedfor each rank separately, the CQI information can be calculated, andthen a decision made as to which rank is optimal. Instead of trying outall of the possible ranks, the exemplary algorithm of FIG. 5, however,starts off with the maximum rank and decreases the rank until the pointwhen the rate supported by the channel does not increase.

FIG. 6 is a flowchart of an exemplary successive beamforming algorithm.At 610, the rank k is provided as an input and at 620, the singularvalue decomposition is performed, as described above. At 630-680, aninner product calculation between the eigenvectors of the channel thecorresponding vectors from the codebook is performed within nestedloops. At 690, the index of the precoding matrix uniquely determined bythe set of quantized vectors is found and is output at 695.

FIG. 7 is a flowchart of an exemplary CQI calculation algorithm for MCWtransmission for reception by an LMMSE decoder. After inputting theprecoding matrix and power allocation vector at 710, an intermediatevariable Z is calculated at 720. A processing loop is then commenced at730 in which a further intermediate variable S is calculated at 740. ACQI for each of the k streams of rank k is then calculated at 750 basedon the intermediate variables S and Z. Operation loops back at 760 untilthe CQIs for all streams have been determined. The CQIs are then outputat 770.

As an alternative to determining a CQI for each stream, the CQIs of allstreams of a rank can be combined, so as to output only one CQI value.

Simulation Results

An exemplary embodiment of the present invention has been simulated withthe parameters indicated in FIG. 8A. A modulation and coding scheme(MCS) table with 32 entries was chosen and was used for CQI quantizationfor all the simulated schemes. Since only two antennas are assumed atthe UE, the transmission rank is at most two. As shown in FIG. 8B, eightbits are used for two-rank precoding feedback. The first bit indicatesthe beamforming rank followed by 7 bits that represents the index of aprecoder for the given rank. For both rank 1 and rank 2 beamforming, the“xxxx” bits represent the feedback bits for the first vector and the“yyy” bits represent the feedback bits for the second vector.

For rank 2, there would be a total of 2⁷ precoding (or beamforming)matrices, comprising the Cartesian product of 2⁴ vectors in C⁴ space and2³ vectors in C³ space. Therefore, the total memory required to storethe codebook would be 16*4+8*3=88 complex numbers. For rank 1,successive beamforming is applied in order to reduce the memoryrequirement from 2⁷*4=512 complex numbers to only 24 complex numbers.

In accordance with the Spatial Channel Model (SCM) defined in 3GPP, a UEwill be dropped with a given distribution when it is between 45 and 500meters from the BS. As such, the initial pathloss and shadowing valuesaffect the entire simulation results per run. In order to average outthe effect of these parameters, the average performance of each schemeover multiple drops was simulated.

As mentioned, the feedback signals are sent per chunk in order to reducethe feedback requirement. For the given simulation setup of FIG. 8A,using 21 adjacent tones was determined to be the best chunk size. FIG. 9illustrates the effect of chunk size on the throughput of the exemplaryscheme at P/N₀=20 dB when the chunk size varies from 1 to 51. P/N₀ isthe ratio of the transmit power to the noise variance. It is observedthat the throughput increases almost linearly at the beginning, with arelatively steep slope and then flattens out with a shallower slope atlarger chunk sizes. In order to reduce the feedback requirement, it isdesirable to increase the chunk size as much as possible, while thethroughput will be adversely affected with larger chunk sizes. FIG. 9shows that choosing approximately 21 tones per chunk provides thehighest ratio of relative throughput per chunk size for the simulatedexemplary embodiment.

FIG. 10 illustrates the relative performance of the simulated exemplaryscheme against fundamental limits using a finite set of rates (only 5CQI bits). FIG. 10 shows that the relative gain due to the addition ofchannel state information at the transmitter (i.e., CSIRT versus CSIR)for the simulated scenario (4 transmit and 2 receive antennas) isconsiderable. Therefore, the importance of an effective feedbackstrategy becomes more dominant. FIG. 10 also shows that the simulatedexemplary feedback strategy (in a typical scenario) is capable ofbridging almost 80% of the gap between the performances of a system withCSIRT versus CSIR only.

FIG. 11 provides a cross comparison of an exemplary scheme in accordancewith the present invention against selected other schemes including: (i)PU2RC by Samsung with 8-bit feedback, (ii) antenna selection with 3-bitsfeedback, (iii) rank adaptation and cycling by Qualcomm, with SCW and1-bit feedback, and (iv) a 2-streams antenna cycling scheme with nofeedback. It should be noted that a thorough comparison betweendifferent schemes can be performed with system level simulation, whereasFIG. 11 provides a link level comparison.

FIG. 12 shows the effect of a typical feedback error on the performanceof the exemplary scheme. It seems that the effect of feedback error onthe rank information bit is more destructive than its effect on theother feedback bits. Therefore, more protection for this bit isindicated.

UE Operation for Alternative Embodiment

FIGS. 13A and 13B show a flowchart which illustrates the UE operationfor an alternative exemplary embodiment of a rank adaptation scheme fora MIMO downlink in which the beamforming rank may be provided by thebase station (BS) and which can accommodate various receivers, includingLMMSE, MMSE-SIC, and ML receivers.

As shown in FIG. 13A, the UE estimates the channel matrix H, thetransmit power, and noise variance at 1310. At 1315, a determination ismade as to whether or not the rank is provided by the BS. If not,operation proceeds to step 1320, in which the UE determines the rank kand power allocation P_(i) using a modified capacity measure f(H). Analgorithm for carrying out step 1320 is described above in greaterdetail with reference to FIG. 5.

Operation proceeds to step 1322 in which the UE determines the precodingmatrix U_(i) using successive beamforming. A successive beamformingalgorithm is described above with reference to FIG. 6.

At 1330, the UE picks the index of the augmented precoding matrix, theaugmented precoding matrix comprising by the precoding matrix U_(i) andthe power allocation P_(i).

If it was determined at 1315 that the rank is provided by the BS,operation proceeds to step 1324 in which the UE determines the precodingmatrix U_(i) using successive beamforming. The successive beamformingalgorithm described above with reference to FIG. 6 can be used for thispurpose. Operation then proceeds to step 1326 in which the quantizedpower allocation that maximizes the following expression is determined:

log det[I+PHU diag(P _(i))U ^(H) H ^(H)]  (23)

This expression defines the achievable rate of a precoded MIMO systemwith precoding matrix U and power allocation given by diag(P_(i)),assuming a Gaussian codebook.

The indexing is done by scanning row-by-row or column-by-column theblack-and-white image representing UV spots and assigning a number toeach spot in order.

It is understood that the above-described embodiments are illustrativeof only a few of the possible specific embodiments which can representapplications of the invention. Numerous and varied other arrangementscan be made by those skilled in the art without departing from thespirit and scope of the invention.

1. A transmitter used in a wireless communications system, thetransmitter comprising: a precoding unit to precode data with aprecoding matrix selected from a codebook comprising matrices that areexpressed as${V = \left\lbrack {v^{1},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\v^{2}\end{bmatrix}}},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\{{\Phi \left( v_{1}^{2} \right)}{{{HH}\left( {\frac{v^{2}}{\Phi \left( v_{1}^{2} \right)} - e_{1}^{M - 1}} \right)}\begin{bmatrix}0 \\v^{3}\end{bmatrix}}}\end{bmatrix}}},\ldots} \right\rbrack},$ where the transmitter has Mtransmit antennas, e₁ ^(N)=[1, 0, . . . , 0]^(T)εC^(N) (N=M, M−1, . . .), C^(N) being N-dimensional complex space, v^(a) denotes a columnvector, v_(b) ^(a) denotes the b^(th) element of vector v^(a),${{\Phi \left( v_{b}^{a} \right)} = \frac{v_{b}^{a}}{\left. ||v_{b}^{a} \right.||}},$and ${{HH}(w)} = \left\{ \begin{matrix}{I - {2\frac{{ww}^{H}}{\left. ||w \right.||^{2}}}} & {{{if}\mspace{14mu} w} \neq 0} \\I & {{{if}\mspace{14mu} w} = 0}\end{matrix} \right.$ is the house-holder transformation of vector w;and a transmission unit to transmit the precoded data to a receiver. 2.The transmitter as in claim 1, wherein the transmitter comprises a basestation and the receiver comprises a user equipment.
 3. The transmitteras in claim 1, wherein the transmission is downlink transmission.
 4. Thetransmitter as in claim 1, wherein v^(a) is selected from a set V^(a)whereV ¹ ={v _(i) ¹ εC ^(M)}_(i=1) ^(N) ¹ , V ² ={e ₁ ^(M-1) }, V ³ ={e ₁^(M-2)}, . . . .
 5. The transmitter as in claim 1, wherein the number oftransmit antennas is
 2. 6. The transmitter as in claim 1, wherein thenumber of transmit antennas is
 4. 7. The transmitter as in claim 1,wherein the precoding matrix is formed by selecting one or more columnsof matrix V.
 8. The transmitter as in claim 7, wherein the number ofselected columns to form the precoding matrix is less than the number oftransmit antennas.
 9. A receiver used in a wireless communicationssystem, the receiver comprising: a receiving unit to receive, from atransmitter, data precoded with a precoding matrix selected from acodebook comprising matrices that are expressed as${V = \left\lbrack {v^{1},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\v^{2}\end{bmatrix}}},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\{{\Phi \left( v_{1}^{2} \right)}{{{HH}\left( {\frac{v^{2}}{\Phi \left( v_{1}^{2} \right)} - e_{1}^{M - 1}} \right)}\begin{bmatrix}0 \\v^{3}\end{bmatrix}}}\end{bmatrix}}},\ldots} \right\rbrack},$ where the transmitter has Mtransmit antennas, e₁ ^(N)=[1, 0, . . . , 0]^(T)εC^(N) (N=M, M−1, . . .), C^(N) being N-dimensional complex space, v^(a) denotes a columnvector, v_(b) ^(a) denotes the b^(th) element of vector v^(a),${{\Phi \left( v_{b}^{a} \right)} = \frac{v_{b}^{a}}{\left. ||v_{b}^{a} \right.||}},$and ${{HH}(w)} = \left\{ \begin{matrix}{I - {2\frac{{ww}^{H}}{\left. ||w \right.||^{2}}}} & {{{if}\mspace{14mu} w} \neq 0} \\I & {{{if}\mspace{14mu} w} = 0}\end{matrix} \right.$ is the house-holder transformation of vector w.10. The receiver as in claim 9, wherein the transmitter comprises a basestation and the receiver comprises a user equipment.
 11. The receiver asin claim 9, wherein the reception is downlink reception.
 12. Thereceiver as in claim 9, wherein v^(a) is selected from a set V^(a) whereV ¹ ={v _(i) ¹ εC ^(M)}_(i=1) ^(N) ¹ , V ² ={e ₁ ^(M-1) }, V ³ ={e ₁^(M-2)}, . . . .
 13. The receiver as in claim 9, wherein the number oftransmit antennas is
 2. 14. The receiver as in claim 9, wherein thenumber of transmit antennas is
 4. 15. The receiver as in claim 9,wherein the precoding matrix is formed by selecting one or more columnsof matrix V.
 16. The receiver as in claim 15, wherein the number ofselected columns to form the precoding matrix is less than the number oftransmit antennas.
 17. A receiver used in a wireless communicationssystem, the receiver comprising: a feedback unit to provide feedbackinformation for transmission in order for a transmitter to select aprecoding matrix from a codebook comprising matrices that are expressedas${V = \left\lbrack {v^{1},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\v^{2}\end{bmatrix}}},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\{{\Phi \left( v_{1}^{2} \right)}{{{HH}\left( {\frac{v^{2}}{\Phi \left( v_{1}^{2} \right)} - e_{1}^{M - 1}} \right)}\begin{bmatrix}0 \\v^{3}\end{bmatrix}}}\end{bmatrix}}},\ldots} \right\rbrack},$ where the transmitter has Mtransmit antennas, e₁ ^(N)=[1, 0, . . . , 0]^(T)εC^(N) (N=M, M−1, . . .), C^(N) being N-dimensional complex space, v^(a) denotes a columnvector, v_(b) ^(a) denotes the b^(th) element of vector v^(a),${{\Phi \left( v_{b}^{a} \right)} = \frac{v_{b}^{a}}{\left. ||v_{b}^{a} \right.||}},$and ${{HH}(w)} = \left\{ \begin{matrix}{I - {2\frac{{ww}^{H}}{\left. ||w \right.||^{2}}}} & {{{if}\mspace{14mu} w} \neq 0} \\I & {{{if}\mspace{14mu} w} = 0}\end{matrix} \right.$ is the house-holder transformation of vector w.18. The receiver as in claim 17, wherein the transmitter comprises abase station and the receiver comprises a user equipment.
 19. Thereceiver as in claim 17, wherein the transmission is downlinktransmission.
 20. The receiver as in claim 17, wherein v^(a) is selectedfrom a set V^(a) whereV ¹ ={v _(i) ¹ εC ^(M)}_(i=1) ^(N) ¹ , V ² ={e ₁ ^(M-1) }, V ³ ={e _(i)^(M-2)}, . . . .
 21. The receiver as in claim 17, wherein the number oftransmit antennas is
 2. 22. The receiver as in claim 17, wherein thenumber of transmit antennas is
 4. 23. The receiver as in claim 17,wherein the precoding matrix is formed by selecting one or more columnsof matrix V.
 24. The receiver as in claim 23, wherein the number ofselected columns to form the precoding matrix is less than the number oftransmit antennas.
 25. The receiver as in claim 17, wherein the feedbackinformation includes at least one of a rank, a precoding matrix index,quantization power control, and a Signal to Interference and Noise Ratio(SINR).
 26. A wireless communications system comprising: a transmitterto precode data with a precoding matrix selected from a codebookcomprising matrices that are expressed as${V = \left\lbrack {v^{1},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\v^{2}\end{bmatrix}}},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\{{\Phi \left( v_{1}^{2} \right)}{{{HH}\left( {\frac{v^{2}}{\Phi \left( v_{1}^{2} \right)} - e_{1}^{M - 1}} \right)}\begin{bmatrix}0 \\v^{3}\end{bmatrix}}}\end{bmatrix}}},\ldots} \right\rbrack},$ where the transmitter has Mtransmit antennas, e₁ ^(N)=[1, 0, . . . , 0]^(T)εC^(N) (N=M, M−1, . . .), C^(N) being N-dimensional complex space, v^(a) denotes a columnvector, v_(b) ^(a) denotes the b^(th) element of vector v^(a),${{\Phi \left( v_{b}^{a} \right)} = \frac{v_{b}^{a}}{\left. ||v_{b}^{a} \right.||}},$and ${{HH}(w)} = \left\{ \begin{matrix}{I - {2\frac{{ww}^{H}}{\left. ||w \right.||^{2}}}} & {{{if}\mspace{14mu} w} \neq 0} \\I & {{{if}\mspace{14mu} w} = 0}\end{matrix} \right.$ is the house-holder transformation of vector w;and a receiver to receive the precoded data from the transmitter. 27.The wireless communications system as in claim 26, wherein thetransmitter comprises a base station and the receiver comprises a userequipment.
 28. The wireless communications system as in claim 26,wherein the transmission is downlink transmission.
 29. A wirelesscommunications system comprising: a transmitter to receive feedbackinformation for transmission; and a receiver to provide the feedbackinformation in order for the transmitter to select a precoding matrixfrom a codebook comprising matrices that are expressed as${V = \left\lbrack {v^{1},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\v^{2}\end{bmatrix}}},{{\Phi \left( v_{1}^{1} \right)}{{{HH}\left( {\frac{v^{1}}{\Phi \left( v_{1}^{1} \right)} - e_{1}^{M}} \right)}\begin{bmatrix}0 \\{{\Phi \left( v_{1}^{2} \right)}{{{HH}\left( {\frac{v^{2}}{\Phi \left( v_{1}^{2} \right)} - e_{1}^{M - 1}} \right)}\begin{bmatrix}0 \\v^{3}\end{bmatrix}}}\end{bmatrix}}},\ldots} \right\rbrack},$ where the transmitter has Mtransmit antennas, e_(i) ^(N)=[1, 0, . . . , 0]^(T)εC^(N) (N=M, M−1, . .. ), C^(N) being N-dimensional complex space, v^(a) denotes a columnvector, v_(b) ^(a) denotes the b^(th) element of vector v^(a),${{\Phi \left( v_{b}^{a} \right)} = \frac{v_{b}^{a}}{\left. ||v_{b}^{a} \right.||}},$and ${{HH}(w)} = \left\{ \begin{matrix}{I - {2\frac{{ww}^{H}}{\left. ||w \right.||^{2}}}} & {{{if}\mspace{14mu} w} \neq 0} \\I & {{{if}\mspace{14mu} w} = 0}\end{matrix} \right.$ is the house-holder transformation of vector w.30. The wireless communications system as in claim 29, wherein thetransmitter comprises a transmitter and the receiver comprises areceiver.
 31. The wireless communications system as in claim 29, whereinthe transmission is downlink transmission.